# Inverse Random Source Problem for the Helmholtz Equation from Statistical Phaseless Data

## TL;DR

This paper develops a novel method to reconstruct and analyze a random source for the Helmholtz equation using statistical phaseless data, overcoming non-uniqueness with phase retrieval and Bayesian techniques.

## Contribution

It introduces a reference source technique, phase retrieval formulas, and Bayesian reconstruction for the inverse random source problem, providing theoretical analysis and numerical validation.

## Key findings

- Successfully reconstructs the expectation and variance of the random source.
- Proves stability and uniqueness of the inverse problem solutions.
- Demonstrates effectiveness through numerical experiments.

## Abstract

This paper investigates the problem of reconstructing a random source from statistical phaseless data for the two-dimensional Helmholtz equation. The major challenge of this problem is non-uniqueness, which we overcome through a reference source technique. Firstly, we introduce some artificially added point sources into the inverse random source system and derive phase retrieval (PR) formulas for the expectation and variance of the radiated fields. This paper rigorously analyze the uniqueness and stability of the recovered statistics of the radiated fields. Afterwards, since the direct problem has a unique mild solution, by examining the expectation and variance of this solution and combined with the phase retrieval formulas, we derive the Fredholm integral equations to solve the inverse random source problem (IRSP). We prove the stability of the corresponding integral equations. To quantify the uncertainty of the random source, we utilize the Bayesian method to reconstruct the random source and establish the well-posedness of the posterior distribution. Finally, numerical experiments demonstrate the effectiveness of the proposed method and validate the theoretical results.